Numerical computation of 3D Brownian motion of thin film nanofluid flow of convective heat transfer over a stretchable rotating surface

This research examines the thin-film nanomaterial movement in three dimensions over a stretchable rotating inclined surface. Similarity variables are used to transform fundamental systems of equations into a set of first-order differential equations. The Runge–Kutta Fourth Order approach is utilized for numerical computations. The impact of embedded parameters (variable thickness, unsteadiness, Prandtl number, Schmidt number, Brownian-motion, and thermophoretic) is examined carefully. Physically and statistically, the indispensable terms namely Nusselt and Sherwood numbers are also investigated. Results indicated that, as the dimensionless parameter S raises, the temperature field decreases. In reality, as the values of S increases, heat transmission rate from the disc to the flowing fluid reduces. Internal collisions of liquid particles are physically hampered at a low rate. The momentum boundary layer is cooled when the parameter S is increased, as a consequence local Nusselt number rises. Sherwood number decreases as the parameter S increases because of inter collision of the microscopic fluid particles. Enhancing in the apparent viscosity and concentrations of the chemical reactions, a higher Schmidt number, Sc, lowers the Sherwood number. With increasing values of Prandtl number the Nusselt number decreases. For validation purpose, the RK4 method is also compared with homotopy analysis method (HAM). The results are further verified by establishing an excellent agreement with published data.

In the sphere of chemistry and applied sciences, the development of liquid condensate from a cool, saturated vapor is crucial. Many researchers have looked into this phenomenon under a variety of circumstances. Gregg et al. 1 used the centrifugal force characteristic on a cold spinning disc to investigate the removal of moisture. They converted the fundamental flow equations into highly nonlinear equations and attempted the numerical solution for liquid layer thicknesses of finite and varying thickness. Hudson et al. 2 carried their work a step further by incorporating vapour drag. The theory of Sarma et al. 3 has been expanded to include the adhesion term at the plate surface.
The mutual fluid, which has a poor thermal conductivity, is employed as a basis fluid in much of the available literature. The outputs of these types of heat systems are extremely low. Nanoparticles are tiny particles that are injected inside a base fluid to improve the chemical property of consideration fluid. Hatami 4 investigated the discharge of a nanomaterial across a revolving, inclined plane. Significant physical results for cooling purposes were preserved.
The application of time-dependent flow field in engineering and physical science is equally significant. In situation of porous medium, Attia 5  www.nature.com/scientificreports/ of such sheets, the problems in molten state are stretched from either a gap to reach the appropriate size. The finished product mostly with needed qualities is manufactured due to temperature stretching rate throughout the procedure as well as the stretching cycle. Abu 40 investigated Brownian motion and thermal flexibility impacts on MHD viscoelastic moment of a nanomaterial via a stretchable porous material. Alim et al. 41 examine the MHD time dependent motion of a nanoliquid through a longitudinal stretch sheet under suction/injection.There's been enough investigation on displacement past stretched surfaces. Khan 42 was the first one to address flow on smooth and substantial continuous surfaces. Gul et al. 43 looked at the fluid flow an extendable barrier by assuming that only the surface velocity varied linearly from the slit. MHD solution of a viscoelastic-non-Newtonian liquidover a stretching disc was discovered by Gul et al. 44 . Aziz et al. 45 studied the impact of varied Al2O3 water Nano liquid characteristics on the enhancement of heat transmission in entropy generation. Khan et al. 46 studied heat generation using a flat-plate absorber and a nanofluids with changeable characteristics. In study, thin fluids film fluxes are receiving a lot of attention. Variable parameters of a thin fluid flow through an extending region were studied by Qasim et al. 47 . Prashant et al. 48 intensively reviewed the flow as well as heat transmission non-Newtonian liquid via a porous media through shrinking surface. Gireesha et al. 49 53 demonstrated heat transfer over magnetic field using time-dependent stretch sheet along thermal radiation of thin flow. The flow phenomena and heat transmission past small cylinders has undergone a significant revolution in recent years. The unavoidable needs for slim devices that reduce drag while delivering whole lift in order to keep the body afloat in specific scenarios. In narrow cylinder, radius can be like the boundary layer thickness having is axisymmetric behavior rather than two-dimensional, and the governing equation includes a transverse curve that impacts the temperature and velocity fields through force. A normal curvature has an effect on coefficient of skin friction and rate of heat transfer at the wall, which is relevant to this concept. The preparation of combustion chambers, chimney stacks, coolers, offshore structures, thin film deposition, and paper manufacture are all examples of flow past a cylinder and related transfer of heat properties. Sheikoleslami et al. 54 was the one to evaluate the third-grade non-linear viscous fluid flow via a stretched circular tube. Ahmad et al. 55 concentrate the fluid motion beside a extending tube utilizing Kellerbox approach for solution. The analogous results of the natural convection investigation over a quasi-stretched cylinder were obtained by Sheikholeslami 56 . Wang 57 provided a computational solution of MHD Newtonian fluid moment through a stretched disc. Nanomaterial mobility with heat as well as attractive field over extending surface was described by Kleinstreuer et al. 58  The goal of this work is to investigate the spraying nanomaterial fluid across an angled rotating disc for cooling purposes, in light of the preceding critical debate. Through suitable transformations, the basic equations of continuity, momentum, thermal boundary layer, as well as mass for time dependent density flow are rehabilitated to non-linear ordinary differential equations (ODEs). To generate first order ODEs, these are additionally distorted in order to obtain numerical solution. The numerical solutions of the transformed first order ODEs were achieved using the RK4 technique. The numerical results are indeed validated using the HAM for the sake of confirmation. Furthermore, we verified the acquired results by establishing a comparing with previous literatures, and we discovered an outstanding match, confirming the accuracy of the current communication.

Modeling of the problem
Take a rotating disc with a 3D unsteady nanomaterial thin-film moment. As seen in Fig. 1, the disc rotates with angle . The horizontal line has been at an inclination β with the slanted disc. The nanomaterial sheet thickness is denoted by h , as well as the spray speed is indicated by W . Because the fluid film's thickness is already so thin in comparison to the radius of the disc, the terminal effect is neglected. The gravitation force g is exerting in the negative direction as it often does. The temperature θ 0 is at the film surface, whereas θ w is over the disc. The Concentration happening on surface film is C 0 , while the concentration on surface of is C h .

Result and discussion
The heat and mass transfer across an unsteady rotational inclined plane using 3D thin-film nanomaterial flow has been investigated. The findings were acquired using the numerical approach Runge-Kutta fourth order method (RK4), while the analytical solution for the validation purposes is obtained using HAM. We used Δr = 0.001 as that of the scale factor and 10-6 and δ = 2, as the resolution threshold during our computation which gives four decimal places accuracy. Figure 1 depicts the current problem physical configuration. A Figures 2, 3, 4, 5 displays the impact of S on axial as well as radial velocities, drainage moment, and induced moment, respectively. The variation in fluid moment is depicted by increased quantities of said unsteadiness factor S.
For greater values of unsteadiness factor S, the momentum thickness grows, and as a consequence, most of the said kinds of fluid flow fall, as seen in the depicted graphs. Figure 6 illustrates that the temperature distribution becomes substantially decreaseswith the increasing values of parameter S. In reality, as the values of S increases, heat transmission rate from the disc to the flowing fluid reduces. Internal collisions of liquid particles are physically hampered at a low rate. Because as unsteadiness factor S is increased, the boundary layer momentum increases, as a consequence, the concentration field also enhances, as seen in Fig. 7. (29) y 1 = 0, y 2 = 0, y 3 = 0, y 4 = u 1 , y 5 = 1, y 6 = u 2 , y 7 = 0, y 8 = u 3 , y 9 = 0, y 10 = u 4 , y 11 = 0, y 12 = u 5 , y 13 = 0, y 14 = u 6 .  www.nature.com/scientificreports/ The Nusselt number is a non-dimensional number that describes the relation of thermal energy convected towards the liquid to heat energy conducted inside the medium. The Nusselt value is a measurement of heat transfer rate at the barrier that is equivalent to the non-dimensional temperature difference at the surface. Figure 8 depicts the variation of Nusselt number effects by unsteadiness factor S. It is clear from Fig. 8 that the momentum boundary layer is cooled when the parameter S is increased, as a consequence local Nusselt number rises. The Sherwood number is often used to investigate concentration polarization. The Sherwood number is a non-dimensional number used during mass-transfer operations. It is also known as the mass transfer Nusselt number. In mass-transfer operations, the Sherwood number is a non-dimensional number. It is the proportion   www.nature.com/scientificreports/ of convective mass transfer to diffusive mass transport rate. As illustrated from Fig. 9, that Sherwood number decreases as the parameter S increases because of inter collision of the microscopic fluid particles. As depicted in Fig. 10, large amount of Nt and Nb, enhances heat transfer rate. Nanoparticle movement in nanofluids is caused by thermophoresis and Brownian motion; both have significant influence on the thermo physical properties of nanofluids. The ability of smaller nanostructures to collect at the heated wall and increase the heat transmission rate is demonstrated. Indeed, higher Brownian motion factor Nb upsurges the thickness of thermal boundary layer. With increasing Nb, the stochastic collision among nanoparticles and liquid molecules increases, causing a   www.nature.com/scientificreports/ flow to become heated. Figure 11 demonstrates that how concentration rate decreases with changing of Schmidt number Sc. In fact, increasing the Sc parameter enhances kinematic viscosity and increases chemical species concentration, lowering the Sherwood number. The Prandtl number, also known as the Prandtl group, is a non-dimensional number that represents the ratio of momentum to thermal diffusivity. It is a non-dimensional factor equal to c p μ/k used in thermal performance computations between a fluid moving and a substantial body, where c p the fluid's specific heat in unit volume, μ is the kinematic viscosity, and k is its thermal conductivity. Figure 12 depicts the effect of Pr (Prandtl number), on the heat flux. Thermal boundary layer thickness reduces with enhance of Pr , and so as a consequence, the cooling rate is decreased. The graphical comparison of the RK4 and HAM methods are sketched in Figs. 13, 14, 15 and 16 for the axial and drainage velocities, temperature and concentration fields, respectively, an excellent agreement is noted. Furthermore, the numerical results of the RK4 and HAM methods for the Nusselt number and Sherwood number are given in Table 1. A comparison of the present results with published data is made in limiting sense (see Table 2) which confirms the accuracy and the fact that these results are more general form those in published literature. The Newtonian fluid of the present work can also be obtained by taking S = 0.2, Pr = 6.2, Nt = Nb = Sc = 0andb → 0 as shown in Table 3.

Conclusion
The existing literature focuses primarily on two-dimensional flow problems. The pouring of 3D nanomaterial's across a stretchable inclined rotatable frame is investigated in this paper. The following is a summary of the new findings in the Numerical and analytical solutions: • As the dimensionless parameter S raises, the temperature field decreases.In reality, as the values of S increases, heat transmission rate from the disc to the flowing fluid reduces. Internal collisions of liquid particles are physically hampered at a low rate. • The momentum boundary layer is cooled when the parameter S is increased, as a consequence local Nusselt number rises.         4 for −θ ′ (0) and −φ ′ (0) fixing Pr = 6.5, Nt = Nb = Sc = S = 0.9.S = 0.6. η RK4 −θ ′ (0) Ref. 4 −θ ′ (0)